To solve the quadratic equation [tex]\( y = 3x^2 - 16x - 12 \)[/tex], we need to find the values of [tex]\( x \)[/tex] that make [tex]\( y = 0 \)[/tex]. Essentially, we're looking for the roots of the equation:
[tex]\[ 3x^2 - 16x - 12 = 0 \][/tex]
Here's a step-by-step method you can use to solve this quadratic equation:
1. Identify the coefficients:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = -16 \)[/tex]
- [tex]\( c = -12 \)[/tex]
2. Use the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the discriminant:
The discriminant ([tex]\( \Delta \)[/tex]) is the part of the quadratic formula under the square root:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = (-16)^2 - 4 \cdot 3 \cdot (-12)
\][/tex]
[tex]\[
\Delta = 256 + 144
\][/tex]
[tex]\[
\Delta = 400
\][/tex]
4. Compute the roots:
Substitute [tex]\( \Delta \)[/tex] and the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] back into the quadratic formula:
[tex]\[
x = \frac{-(-16) \pm \sqrt{400}}{2 \cdot 3}
\][/tex]
[tex]\[
x = \frac{16 \pm 20}{6}
\][/tex]
This gives us two solutions:
[tex]\[
x_1 = \frac{16 + 20}{6} = \frac{36}{6} = 6
\][/tex]
[tex]\[
x_2 = \frac{16 - 20}{6} = \frac{-4}{6} = -\frac{2}{3}
\][/tex]
Therefore, the roots of the equation [tex]\( 3x^2 - 16x - 12 = 0 \)[/tex] are:
[tex]\[
x = 6 \quad \text{and} \quad x = -\frac{2}{3}
\][/tex]
